Viewing the spokes

From	Sergio <invalid@invalid.com>
Newsgroups	sci.physics
Subject	Viewing the spokes
Date	2016-08-29 12:08 -0500
Organization	Aioe.org NNTP Server
Message-ID	<nq1q63$bdd$1@gioia.aioe.org> (permalink)

Show all headers | View raw

Viewing the spokes

A wheel with spokes rolls on the ground. A stationary camera takes a 
picture of the wheel. Due to the nonzero exposure time of the camera, 
the spokes will generally appear blurred. At what location(s) in the 
picture does (do) the spoke(s) not appear
blurred?

























answer,


The contact point on the ground does not look blurred, because it is 
instantaneously
at rest. But although this is the only point on the wheel that is at 
rest, there will
be other locations in the picture where the spokes do not appear blurred.
The relevant property of a point in the picture where a spoke does not 
appear
blurred is that the point lies on the spoke throughout the duration of 
the camera’s
exposure. (The point in the picture need not, however, actually 
correspond to the
same atom on the spoke.)
At a given time, consider a spoke in the lower half of the wheel. A 
short time
later, the spoke will have moved (via both translation and rotation), 
but it will
intersect its original position. The spoke will not appear blurred at 
this intersection
point. We must therefore find the locus of these intersections. We can 
do this in
two ways.
First method: Let R be the radius of the wheel. Consider a spoke that 
makes an
angle of µ with the vertical at a given time. If the wheel then rolls 
through a small
angle dµ, the center moves a distance Rdµ. The spoke’s motion is a 
combination
of a translation through this distance Rdµ, plus a rotation through the 
angle dµ
(around the top end).
Let r be the radial position of the intersection of the initial and 
final positions of
the spoke. In the following figure, we can write down two expressions 
for the short
segment drawn perpendicular to the initial position of the spoke.

Rdq
dq
q
q
r
The two expression are (Rdµ) cos µ, and r dµ (to first order in dµ). 
Equating these
gives r = Rcos µ. This describes a circle whose diameter is the lower 
vertical radius
of the wheel, as shown below.
1
q
r=Rcosq
wheel
Second method: Since the wheel’s contact point with the ground is 
instantaneously
at rest, the wheel may be considered to be instantaneously rotating around
this point. This means that every atom in the wheel (both in the spokes 
and the
rim) instantaneously traces out the arc of a circle centered at the 
contact point. For
the point P in the figure below, this circle is the dotted one drawn 
(with Q being
the contact point). A spoke will not appear blurred at the point where 
this circular
motion is along the direction of the spoke. That is, a spoke will not 
appear blurred
at the point where the dotted circle is tangent to the spoke, as shown.
R
q
P wheel
O
QWe are therefore concerned with the locus of all points P such that the 
segments
PQ and PO are perpendicular. As seen in the previous diagram above, this 
locus
is the circle whose diameter is the lower vertical radius of the wheel.
2

Back to sci.physics | Previous | Next — Next in thread | Find similar | Unroll thread

Thread

Viewing the spokes Sergio <invalid@invalid.com> - 2016-08-29 12:08 -0500
  Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-29 18:00 +0000
    Re: Viewing the spokes benj <benj@nobody.net> - 2016-08-29 21:00 -0400
      Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-30 01:14 +0000
        Re: Viewing the spokes benj <benj@nobody.net> - 2016-08-29 22:45 -0400
          Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-30 04:14 +0000
            Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 21:59 -0700
              Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-30 05:16 +0000
                Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 23:00 -0700
    Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-08-29 18:29 -0700
  Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 18:29 -0700
  Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 18:46 -0700
    Re: Viewing the spokes benj <benj@nobody.net> - 2016-08-30 19:40 -0400
      Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-30 22:07 -0700
        Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-30 23:48 -0700
          Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-31 09:54 -0700
            Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-09-02 10:00 -0700
              Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-02 13:23 -0700
              Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 11:30 -0700
                Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 13:18 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 14:57 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 16:01 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-04 12:46 -0700
                Re: Red, Blue and Green Ergios <invalid@invalid.com> - 2016-09-07 12:32 -0500
                Re: Red, Blue and Green noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-08 12:12 -0700
                Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-07 23:22 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 16:03 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 16:06 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-20 00:18 -0700
                Re: Red, Blue and Green noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-20 12:52 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-07 10:01 -0700
                Re: Red, Blue and Green Gutless Umbrella Carrying Sissy <taustinca@gmail.com> - 2016-09-07 10:12 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-20 09:47 -0700
        Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-08-31 14:32 -0700
  Re: Viewing the spokes "reber g=emc^2" <herbertglazier0@gmail.com> - 2016-09-04 11:39 -0700
    Re: Viewing the spokes "hanson" <hanson@quick.net> - 2016-09-04 20:36 -0700
  Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-09 09:08 -0700
    Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-09 12:32 -0700
    Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-09 17:51 -0400
      Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-10 00:10 +0000
        Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-09 20:43 -0400
          Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-10 00:57 +0000
            Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-09 21:07 -0400
              Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-10 01:33 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-12 20:59 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-13 02:41 +0000
      Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-09 19:43 -0700
        Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-12 21:29 -0400
          Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-13 02:43 +0000
          Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-13 13:28 -0700
          Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-13 17:14 -0700
            Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-13 21:52 -0400
              Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-14 05:11 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-15 17:49 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 00:50 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-15 21:52 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 04:11 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-16 16:58 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 21:40 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-16 20:56 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-17 01:47 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-17 17:24 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-18 00:07 +0000
              Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-14 14:26 -0700
                Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-14 15:04 -0700
                Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-16 12:54 -0700
              Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-14 19:47 -0700
                Re: Viewing the spokes benj <nobody@gmail.com> - 2016-09-14 23:24 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-15 05:12 +0000
                Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-15 15:45 -0700
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-15 21:36 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 02:58 +0000
                Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-18 12:15 -0700
    Re: Viewing the spokes Wally W. <ww84wa@aim.com> - 2016-09-15 09:15 -0400
      Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-15 15:48 -0700
        Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-18 15:33 -0700

csiph-web